I have a B.Sc. in Mechanical engineering, a M.Eng in aeronautical engineering and I am currently looking for a topic for my Ph.D. I've been working in "conventional" CFD for almost 4 years now and I really think it's time to move on beyond the Navier-Stokes equation. I've always been interested in astrophysics, cosmology, and relativity. Relativistic fluid dynamics seems very attractice for the time being. I wonder if anyone may share his thoughts about that topic in terms of how mature is it comparing to mainstream CFD ? is there any open source or commercial code developped ? what are the leading institutes concerned with that research ?

I completed my PhD in Relativistic Magnetohydrodynamics (RMHD), specifically in RMHD of pulsar wind nebulae. Relativistic hydrodynamics (RHD) is a very well developed subject area, with extensive numerical techniques (typically Godunov methods) to deal with theses purely hyperbolic systems (ideal non-dissipative). However, there are limited applications for these relativistic theories away from astrophysics; it is from high energy (and low energy) astrophysical phenomena that we are able to test our relativistic theories.

RHD however, does not incorporate plasma flows which are permeated by magnetic fields and are less applicable to the "sexy" subjects such as black hole accretion, pulsar winds, and other such high energy flows; RHD being more applicable to interstellar clouds, and some varieties of astrophysical jet flows. RMHD is far more applicable to everything astrophysical, however is much more complicated numerically due to the strong and complicated coupling of the equation of state to the Lorentz factor (much more so that RHD). In addition you have to enforce the divergence free condition on the magnetic filed, and the four dimensional nature of relativity means the code has to be fully covariant (especially for general relativistic codes). There are no off the shelf/commercial codes I know of for RHD, these are easily written in one and two-dimensions and I can send you one of mine to check out:

RMHD is a different ball game. There are some amazing codes out there, the 2D Roe-type/like solver by Komissarov (1999) is one of the best and can be used for almost and astrophysical flow. 3D RMHD codes are rare due to there computational expense and the numerical computations that inevitably arise from parallelising and indeed just writing such complicated codes.Leeds, and S. S. Komissarov are a top institute for such studies. Check out his papers and co-others, this will give you the institutes which are funded to undertake such research.

Thank you so much for your informative reply. you've open new doors for me indeed.

I've already started scratching the surface about general relativity, radiation transport and plasma. I've also started making a general literature review on the subject. it seems that I have to learn too much physics before starting but I'm definetly willing to go for it.

Thats no problem at all. If you need any more help reguarding general relativity or RHD/RMHD please don't hesitate to contact me.

Indeed you will have alot of physics to learn but it will be a very rewarding journey, if you have the dedication! Studying compact objects with CFD is a very interesting area and I am sure you will do well if you choose to venture down this road.